11-1
Inverse Variation
TEKS FOCUS
VOCABULARY
ĚCombined variation – a relation
TEKS (6)(L) Formulate and solve equations
involving inverse variation.
ĚJoint variation – a relation in
in which one variable varies with
respect to each of two or more
variables
TEKS (1)(A) Apply mathematics to
problems arising in everyday life, society,
and the workplace.
which one variable varies
directly with respect to each of
two or more variables
ĚInverse variation – a relation
ĚApply – use knowledge or
represented by an equation of
the form xy = k, y = kx , or x = ky ,
where k ≠ 0
Additional TEKS (1)(D), (6)(G), (6)(H)
information for a specific purpose,
such as solving a problem
ESSENTIAL UNDERSTANDING
If a product is constant, where the constant is positive, a decrease in the value of one
factor must accompany an increase in the value of the other factor.
Key Concept
Combined Variations
Combined Variation
Equation Form
z varies jointly with x and y.
z = kxy
z varies jointly with x and y and inversely with w.
z= w
z varies directly with x and inversely with the product wy.
z = wy
kxy
Problem 1
P
kx
TEKS Process Standard (1)(D)
Identifying Direct and Inverse Variations
How can you tell if
the quantities vary
directly or inversely?
If the product of
corresponding x- and
y-values is constant, they
vary inversely. If the ratio
of corresponding x- and
y-values is constant, they
vary directly.
Is the relationship between the variables a direct variation, an inverse variation,
or neither? Write function models for the direct and inverse variations.
o
A
x
y
2
15
4
7.5
10
3
15
2
As x increases, y decreases. This might be an
inverse relationship. A plot confirms that an
inverse relationship is possible. Test to see
whether xy is constant.
# 15 = 30
10 # 3 = 30
2
# 7.5 = 30
15 # 2 = 30
4
The product of each pair is 30, so xy = 30 and y varies inversely with x. The constant
of variation is 30 and the function model is y = 30
x.
continued on next page ▶
450
Lesson 11-1 Inverse Variation
Problem 1
B
Could you model
these data with a
direct variation?
No; the ratio of
corresponding x- and
y-values is not constant.
continued
A plot of the points suggests that an inverse
relationship is possible. Test to see whether
the products of x and y are constant.
x
y
2
10
4
8
10
3
15
1.5
# 10 = 20, 4 # 8 = 32, 10 # 3 = 30, and
15 # 1.5 = 22.5
2
Since the products are not constant, the
relationship is not an inverse variation.
Problem
P
bl
2
Determining an Inverse Variation
Suppose x and y vary inversely, and x = 4 when y = 12.
A What function models the inverse variation?
k
Write the general function form for inverse variation.
k
Substitute for x and y.
y=x
12 = 4
k = 48
Solve for k.
The function is y = 48
x.
Is it reasonable to
connect the points of
this function with a
smooth curve?
Yes, 48
x is defined for
every real number
except x = 0.
B What does the graph of this function look like?
Make a table of values. Sketch a graph.
y
x
y
3
16
4
12
12
6
8
8
8
6
12
16
4
3
16
(3, 16)
(4, 12)
(6, 8)
4
(8, 6)
O
4
8
(12, 4)
(16, 3)
12 16
x
C What is y when x = 10?
48
y= x
Write the function.
48
y = 10
Evaluate y for x = 10.
y = 4.8, when x = 10.
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Problem 3
P
TEKS Process Standard (1)(A)
Modeling an Inverse Variation
Your math class has decided to pick up litter each weekend in a local park. Each
week there is approximately the same amount of litter. The table shows the
number of students who worked each of the first four weeks of the project and
the time needed for the pickup.
Park Cleanup Project
Number of students (n)
Time in minutes (t)
Can you still use
inverse variation to
model the data if
12 × 21 = 252?
Often, you cannot
describe real-life data
exactly with a function
rule. But 252 is close
enough to 255 for
inverse variation to
still be a good model.
3
5
12
17
85
51
21
15
A What function models the data?
Step 1
Investigate the data.
The more students who help, the less time the
cleanup takes. An inverse variation seems
appropriate. If this is an inverse variation, then nt = k.
From the table, nt (or L1 L2) is almost always 255.
#
Step 2
Determine the model. nt = 255
B How many students should there be to complete the project
L1
in at most 30 minutes each week?
nt = 255
n(30) = 255
255
n = 30 = 8.5
3
5
12
17
Use the model from part A.
Substitute for t.
L2
85
51
21
15
L3
3
255
255
252
255
L3=L1 L2
Solve for n.
There should be at least 9 students to do the job in at most 30 minutes.
Problem
bl
4
Using Combined Variation
How can you write
the model?
Write the constant of
variation and direct
variation variable in the
numerator. Write the
inverse variation variable
in the denominator.
Multiple Choice The number of bags of grass seed n needed to reseed a yard
varies directly with the area a to be seeded and inversely with the weight w
of a bag of seed. If it takes two 3-lb bags to seed an area of 3600 ft2, how many
o
3-lb bags will seed 9000 ft2?
3
3 bags
4 bags
6 bags
ka
n varies directly with a and inversely with w.
3600k
3
Substitute for n, a, and w.
n= w
2=
5 bags
(2)(3)
3600 = k
6
1
k = 3600 = 600
Solve for k.
Simplify.
a
The combined variation equation is n = 600w
.
continued on next page ▶
452
Lesson 11-1 Inverse Variation
Problem 4
continued
a
n = 600w
#
9000
= 600 3
=5
Use the combined variation equation.
Substitute for a and w.
You need five 3-lb bags to seed 9000 ft2. The correct choice is C.
Problem
P
bl
5
Applying Combined Variation
STEM
Physics Gravitational potential energy PE is a measure of
energy. PE varies directly with an object’s mass m and its
height h in meters above the ground. Physicists use g to
represent the constant of variation, which is gravity.
The skateboarder in the photo has a mass of 58 kg and a
potential energy of 2273.6 joules. What is the gravitational
potential energy of a 65-kg skateboarder on the halfpipe
shown?
ĚThe mass of each
skateboarder
ĚThe height of each
skateboarder
ĚThe potential energy of the
first skateboarder
The potential energy
of the second
skateboarder
ĚWrite the variation for potential
energy.
ĚUse the known information to
find g.
ĚThen find the potential energy of
the second skateboarder.
Step 1
Write the formula for potential energy. Potential energy
varies directly with mass and height. PE = gmh
Step 2
Use the given data to find g.
PE = gmh
PE
=g
mh
2273.6
=g
(58)(4)
9.8 = g
Step 3
Potential energy formula
Solve for g.
Substitute.
Simplify.
Use the formula to find the potential energy of the
second skateboarder.
PE = 9.8mh
Potential energy formula
= 9.8(65)(4)
Evaluate for m = 65 and h = 4.
= 2548
Simplify.
The second skateboarder has 2548 joules of potential energy.
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453
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Is the relationship between the values in each table a direct variation, an inverse
variation, or neither? Write equations to model the direct and inverse variations.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1.
x
y
3
8
10
22
2.
x
y
15
3
14
40
5
8.4
50
7
6
110
10.5
4
3.
4.
x
y
0.5
1
0.1
3
2.1
4.2
3
0.1
3.5
7
6
0.05
22
24
11
x
y
0.0125
Suppose that x and y vary inversely. Write a function that models each inverse
variation. Graph the function and find y when x = 10.
5. x = 1 when y = 11
6. x = -13 when y = 100
8. x = 2.5 when y = 100
9. x = 5 when y = - 13
7. x = 1.2 when y = 3
4
10. x = - 15
when y = -105
11. Apply Mathematics (1)(A) The number of buckets of paint n needed to paint a
fence varies directly with the total area a of the fence and inversely with the
amount of paint p in a bucket. It takes three 1-gallon buckets of paint to paint
72 square feet of fence. How many 1-gallon buckets will be needed to paint
90 square feet of fence?
12. Apply Mathematics (1)(A) On Earth with a gravitational acceleration g, the
potential energy stored in an object varies directly with its mass m and its
vertical height h.
a. What is an equation that models the potential energy of a 2-kg skateboard
that is sliding down a ramp?
b. The acceleration due to gravity is g = -9.8 m>s 2 . What is the height of the
ramp if the skateboard has a potential energy of -39.2 kg m2 >s 2 ?
13. Apply Mathematics (1)(A) The table shows data about
how the life span s of a mammal relates to its heart
rate r. The data could be modeled by an equation of
the form rs = k. Estimate the life span of a cat with a
heart rate of 126 beats/min.
14. Apply Mathematics (1)(A) The force F of gravity on
a rocket varies directly with its mass m and inversely
with the square of its distance d from Earth. Write a
model for this combined variation. Write an equation
to find the mass of the rocket in terms of F and d.
Heart Rate and Life Span
Heart rate
(beats/min)
Life span
(min)
Mouse
634
1,576,800
Rabbit
158
6,307,200
76
13,140,000
Mammal
Lion
SOURCE: The Handy Science Answer Book
15. Connect Mathematical Ideas (1)(F) Suppose that 1 x1, y1 2
x
y
and 1 x2, y2 2 are values from an inverse variation. Show that x12 = y21.
454
Lesson 11-1 Inverse Variation
16. The spreadsheet shows data that could be modeled by an
equation of the form PV = k. Estimate P when V = 62.
A
B
Write the function that models each variation. Find z when x = 4
and y = 9.
1
P
V
2
140.00
100
17. z varies directly with x and inversely with y. When x = 6 and
y = 2, z = 15.
3
147.30
95
4
155.60
90
5
164.70
85
6
175.00
80
7
186.70
75
18. z varies jointly with x and y. When x = 2 and y = 3, z = 60.
19. z varies inversely with the product of x and y. When x = 2 and
y = 4, z = 0.5.
20. Explain Mathematical Ideas (1)(G) Explain why 0 cannot be
in the domain of an inverse variation.
21. Use Multiple Representations to Communicate Mathematical Ideas (1)(D)
The height h of a cylinder varies directly with its volume V and inversely with
the square of its radius r. Find at least four ways to change the volume and
radius of a cylinder so that its height is quadrupled.
Each ordered pair is from an inverse variation. Find the constant of variation.
22. (6, 3)
24. ( 12, 118)
23. (0.9, 4)
Each pair of values is from an inverse variation. Find the missing value.
25. (2, 5), (4, y)
26. (4, 6), (x, 3)
27. (x, 12), (4, 1.5)
TEXAS Test Practice
T
28. Which equation represents inverse variation between x and y?
y
15y
15z
A. x = z
B. x = - y
29. How can you rewrite the expression (8 F. 39 + 80i
C. z = - x
5i)2
G. 39 - 80i
D. xz = 5y
in the form a + bi?
H. 89 + 80i
J. 89 - 80i
30. The height of a ball thrown straight up from the ground with a velocity of 96 ft>s is
given by the quadratic function h (t) = -16t 2 + 96t. What is the maximum height
the ball reaches?
A. 6 ft
B. 128 ft
C. 144 ft
D. 160 ft
6
31. Which expression is NOT equivalent to 281x4y 8 ?
2
F. 1 3xy 2 2 3
2 4
G. (3x)3y 3
1
H. (3x2y 2)3
3
J. 29x2y 4
32. What is the inverse of y = 4x2 + 5? Is the inverse a function?
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455
Technology Lab
USE WITH LESSON 11-2
Graphing Rational Functions
TEKS (2)(A), (2)(G), (1)(F)
You can use your graphing calculator to graph rational functions and other members of
the reciprocal function family. It is sometimes preferable to use the DOT plotting mode
rather than CONNECTED plotting mode. The CONNECTED mode can join branches of a
graph that should be separated. Try both modes to get the best graph.
Example
4
Graph y = x − 3 − 1.5.
Step 1 Press the mode key. Scroll
down to highlight the word
DOT. Then press enter .
Step 2 Enter the function. Use
parentheses to enter the
denominator accurately.
Plot1
Plot2
\Y1 = 4/(X – 3) –1.5
\Y2 =
\Y3 =
\Y4 =
\Y5 =
\Y6 =
\Y7 =
Normal Sci Eng
F l o a t 0 12 3 4 5 6 7 8 9
Radian Degree
Func Par Pol Seq
Connected Dot
Sequential Simul
R e a l a + b i r e ^ ϴi
F u l l H o r i z G –T
Step 3 Graph the function.
Plot3
Exercises
1. a. Graph the parent reciprocal function y = 1x .
b. Examine both negative and positive values of x. Describe what happens to the
y-values as x approaches zero.
c. What happens to the y-values as x increases? As x decreases?
2. a. Change the mode on your calculator to CONNECTED. Graph the function from
the example.
b. Press trace and trace the function. What happens between x ≈ 2.8 and
x ≈ 3.2?
c. Analyze Mathematical Relationships (1)(F) How does your graph differ from
the graph in the example? Explain the differences.
Use a graphing calculator to graph each function. Then sketch the graph.
3
4. y = x + 4 - 2
5. y =
4x + 1
7. y = x -2 2
8. y = x +1 2 + 3
6. y = x - 3
2x
9. y = x + 3
456
x+2
(x + 1)(x + 3)
7
3. y = x
Technology Lab
10. y =
x2
x2 - 5
Graphing Rational Functions
11. y =
20
x2 + 5